This would imply that gcd(2,10)=1, but gcd(5,10) != 1 so 5 and 10 are not co-prime....

10 = 5 * 2
5 = ( 2 * 2 ) + 1
2 = ( 2 * 1 )

10 = 2 * 5
2 = ( 2 * 1 )

But what I can't get in my head is...

by the definition of co-prime, two numbers m,n are said to be co-prime if they share no proper factors ( i.e gcd(m,n) = 1 ).

For 5 and 10, their greatest factor ( other than itselt 5 ) is 1, since all that go into 10 and 5 properly are 5 & 1, but under the same assumption couldn't you say this for 2 and 10, since the only numbers that go into 2 & 10, are 2 & 1.

This would make them both co-prime, yet this would imply that the 'offending statement' is not true....

I have a funny feeling I am missing something obvious here.... help would be appreciated...

Converse Confusion

In modular arithmetic, exists if and only if and are coprime.

Suppose . Since , there exists an element . Multiplying on both sides gives so .

As for your example, and . This absolutely holds, but that is not to say that one implies the other. Remember basic logic, if there is an earthquake in India and the stock market crashes in America, one did not necessarily imply the other . Similarly AND does not imply . That would be the converse of the above correct theorem, and the converse of a true statement cannot be assumed to be true.

When you are "dividing by two" that is actually an illegal operation in modulo. You are implying that there exists a number , where in fact this number does not exist. Likewise, in your second example, does not exist . Therefore if and where , this can only be attributed to coincidence, not as proof that the converse holds.